The influence of multiplicative noise on the dynamics of active Rab5 and Rab7 proteins on the surface of endosomes has been studied. We have formulated a stochastic model of the conversion of early endosomes into late endosomes involving the replacement of Rab5 by Rab7. This model consists of two stochastic differential equations for protein levels Rab5 and Rab7 on the surface of endosomes. Numerical simulations show that Rab5 experiences a quicker initial growth compared to Rab7, but the latter eventually surpasses Rab5 and becomes dominant with strong random fluctuations. This process leads to a random decrease in endosomal pH and, as a consequence, the random escape of pH-sensitive nanoparticles and viruses from the endosomes. The distribution of pH values in endosomes and the proportion of endosomes with pH lesser than the critical value have been obtained. Our numerical results show that when comparing multiplicative and additive noise, the characteristic time for nanoparticles/viruses to escape from endosomes is shorter with multiplicative noise.